Write out the proof from Example 2.10 that 0 < pi < 1.

Example 2.1. Here S(0) = 5, S(1,) = 20/3 and S(1,) = 40/9. B(0) = 1 and B(1,)=B(1,)=R= 10/9. So r= 1/9 and(2.1)clearlyholds.

Suppose X(1) is any claim that will be paid at time t = 1. In our model X(1) can take one of two values: X(1,) or X(1,). We shall determine X(0), the premium or price of X at time t = 0.

Often the values of X(1) are uncertain because X(1) = f(S(1)) (a function of S) and S(1) is uncertain. As X is an asset whose value depends on S, it is a derived asset written on S, or a derivative on S. X is also called a derivative or a contingent claim.